Study of interactions between close HVDC links inserted in an AC grid: A mixed nonlinear and modal analysis approach

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Study of interactions between close HVDC links

inserted in an AC grid: A mixed nonlinear and modal

analysis approach

Iulian Munteanu, Bogdan Marinescu, Florent Xavier

To cite this version:

Iulian Munteanu, Bogdan Marinescu, Florent Xavier. Study of interactions between close HVDC links

inserted in an AC grid: A mixed nonlinear and modal analysis approach. International Transactions

on Electrical Energy Systems, Wiley, 2019, �10.1002/2050-7038.12266�. �hal-02510539�

Study of interactions between close HVDC links

inserted in an AC grid: A mixed nonlinear and

modal analysis approach

Iulian Munteanu

, Bogdan Marinescu

, Florent Xavier

1 Ecole Centrale Nantes-LS2N, 1 rue de la No¨

e, 44321 Nantes, France

3 R´

eseau de Transport d’´

Electricit´

e, France

2 corresponding author: bogdan.marinescu@ec-nantes.fr

January 3, 2019

Abstract

This paper focuses on interaction between two closed-connected voltage DC (HVDC) lines. This interaction is studied by employing high-fidelity nonlinear modeling in MATLAB R

/Simulink R

software environ-ment. In order to describe the mechanism behind the HVDCs interaction, both nonlinear time-domain simulations and modal analysis of the cou-pled HVDC links, have been performed. System small-signal stability has been assessed and the path of interactions has been identified by com-puting the participations of various states in the oscillatory modes; this sets the preliminaries for a global, grid-oriented HVDCs control design approach. After the detailed analysis of newly-identified coupling oscilla-tions, the minimal modeling requirements to put them into evidence have also been studied; this significantly facilitates the analysis in a realistic large-scale grid context.

Keywords: HVDC interactions, oscillatory modes, minimal modeling re-quirements, small-signal stability

1

Introduction. Scope of the paper

High-Voltage Direct Current (HVDC) lines are more and more used to reinforce transmission capacity of AC grids – [1]. As they are active elements, with multiple control degrees of freedom, their operation are likely to interfere due to geographic proximity (see, for example, the North-South HVDC projects in Germany or the ones on the border France-Spain - [2]), with potentially negative effects on their stability and the one of the neighbor AC zone.

HVDC lines interaction is an important issue for the power systems commu-nity – this is advocated by early publications that concerns HVDCs based upon

line-commutated converters (see, for example, [3]) and stated by international working groups ([4], [5]).

A thorough analysis of interactions between the converters of closed-connected HVDCs immersed in the AC grid is required, in order to properly qualify their closed-loop operating performance. In this sense, parallel voltage-source con-verters (VSC) operation in AC grid context have been studied in papers like [6], [7], where overall system stability is assessed by using converters input-admittance modeling. In [8] analysis of multi-terminal HVDC systems is ap-proached, with focus on the DC system. Interaction modes between different VSCs has been revealed and their mechanism has been assessed. A particular and important situation occurs when the HVDC is inserted in a meshed AC system. In this case, not only interactions between the converters via the DC connections should be taken into account, but also direct connection via the AC grid. This results in dynamic coupling of the voltage/reactive power of the two converters AC connection points. In [9], control solutions are proposed in this context to reduce interaction between two close HVDCs using simplified phasor models.

These works revealed that entire HVDC line modeling at a certain level of detail is required, in order to properly identify and assess the phenomena that lie behind the HVDCs interactions.

The present work first approaches these interactions analysis by using elec-tromagnetic transients (EMT)-based modeling which captures the VSC dy-namic behavior for the full frequency range. Their nature and mechanism are explained via both nonlinear (in simulation) and modal analysis, run in a com-plementary way. Next, it is studied which are the minimal modeling require-ments needed to emphasize the coupling phenomena. A preliminary study of these phenomena has been done in [10].

The approached system structure consists of two two-level HVDCs intercon-nected by means of a common AC power line, further denoted as ConAC – see Fig. 1. This is derived from the benchmark in [4], all AC lines and generators being simplified to impedances and infinite buses. The simplification allows to better identify dynamic properties belonging exclusively to HVDCs coupling. Also, this model is simple enough to be analytically treated in detail; by varying its parameters - especially the lengths of the AC lines and the level of injected power - one can take into account a wide range of realistic coupling situations. Thus, this benchmark is meaningful for studying the AC-DC interactions in the case of HVDC inserted in AC grids.

System behavior is assessed by means of modal analysis for continuous-time linear systems – [11], which offers a tool of determining of all (oscillatory) modes and of the degree of participation of system states in a certain mode. Originally used for stability issues in classical power system analysis, this approach may be applied to other types of complex systems, in which decoupling methods cannot be employed. A similar approach using multimodular converter (MMC)-based HVDC has been briefly presented in [12]. The present work extends system evaluation by thoroughly assessing states influence in shaping HVDC’s interaction for various operating scenarios and parameters of the system. It is

1

1

1

1

Figure 1: General overview concerning the interconnection of two HVDC lines by means of an AC line. Synchronous generator may be present into the AC grid, AC line 1 is an equivalent of a series connection of AC line 5 and AC line 6, in the case where G1 does not exist.

shown that this kind of interaction is of structural type, similar – to some extent – to the well known inter-area modes which exist between rotating generators. These interactions have been captured by means of a model able to replicate with sufficient accuracy transient phenomena within 0.1 Hz – 3 kHz frequency range (low-frequency oscillations, [13]).

The modeling approach in this paper has a twofold goal. It sets the prelim-inaries for a global, grid-oriented HVDCs control design approach and strives to precisely identify the circuit elements whose detailed modeling is relevant to replicate the HVDC-related phenomena in the frequency range of interest. This significantly facilitates the analysis in a realistic large-scale grid context.

Further, the paper is organized as follows. Section 2 goes on with system modeling aspects, while Section 3 presents the overall analysis approach. Sec-tions 4 and 5 present results concerning the HVDCs interaction analysis. Section 6 identifies the minimal modeling required in order to capture the HVDC inter-actions. Section 7 reiterates the HVDC interaction analyses when the AC grid contains controlled synchronous generators. Section 8 concludes the paper.

2

Modeling approach

The approached system from Fig. 1 will be modeled by using the following general assumptions. AC environment is symmetrical and balanced and is mod-eled by AC infinite buses and series reactances. AC-DC converters are lossless and are switching instantaneously, the converters switching dynamics being ne-glected. The connection transformer, usually placed at a certain Point of Com-mon Coupling (PCC), has negligible magnetizing impedance and presents no magnetic saturation phenomena; for sake of analysis, its dynamic behavior and the one of the phase reactor will be lumped into a single RL passive element, the

LDC; rDC Lr1; rr1 Lg1; rg1

1

₁

Figure 2: Simplified diagram of HVDC transmission line and its overall control.

whole AC system becomes rated at a common voltage value like in [10] – this is also suggested in Fig. 2. The interconnection AC line (ConAC in Fig. 1) is modeled as an RL circuit element. Concerning the synchronous generator, the modeling assumptions consider sinusoidal evolutions of voltages and currents in the stator, no magnetic saturation and neglect effects of rotor amortisseurs (only the field circuit is considered in the rotor).

2.1

HVDC structure and assumptions

One considers a symmetrical monopole VSC-HVDC transmission power link whose equivalent diagram (with lumped DC lines and equivalent capacitors) is depicted in Fig. 2. It has two converter stations that employ bidirectional 3-phase AC-DC power converters, interlinked by means of DC cables and con-nected to equivalent AC grids, represented by infinite buses. Subscript indexes of various variables designate the station to which these variables belong.

VSC switched operation takes place at a high frequency with respect to the main dynamics and requires filtering elements on each station: capacitors on DC side and line reactors on AC side.

VSCs of HVDC link are operated in pulse-width modulation (PWM) in order to interchange averaged sinusoidal variables with AC grid. The converter switching dynamics are neglected and VSCs are modeled here by their averaged model, obtained by replacing converter switching functions with their sliding averages – see [14].

2.2

HVDC dq frame modeling

The power subsystem model results from analyzing the electrical interactions between VSC and infinite bus through passive circuit elements. The model is then described in rotating dq frame linked to the infinite bus; on d channel one conveys active power and on q channel reactive power. One writes

dynamic model of HVDC station S1 connected to the equivalent grid, by means of equation set (1):                    Lr1ird1˙ = ωLr1· irq1− 0.5vDC1· βrd1+ vrd1− rr1· ird1 Lr1irq1˙ = −ωLr1· ird1− 0.5vDC1· βrq1+ vrq1− rr1· irq1 CvDC1˙ = 1.5 (ird1· βrd1+ irq1· βrq1) − 2idc Lg1igd1˙ = ωLg1· igq1− vrd1+ E − rg1· igd1 Lg1igq1˙ = −ωLg1· igd1− vrq1− rg1· igq1 vrd1 = (igd1− ird1) · RD vrq1 = (igq1− irq1) · RD, , (1)

where ird1and irq1are the reactor current components, igd1 and igq1are the

AC line current components, vrd1 and vrq1are the PCC voltage components, E

is the infinite bus voltage amplitude, ω is the AC system frequency and vDC1

is the DC line voltage at the concerned VSC (capacitor filter voltage). Lr1

and rr1 are the reactor inductance and resistance, respectively, Lg1 and rg1

are the AC line inductance and resistance, respectively and C1 is the value of

DC-side capacitor. βrd1 and βrq1 are plant (station S1) control inputs. The

first three equations from set (1) actually represent VSC model as is classically described in the concerned bibliography – e.g., [15, 16]. The first two equations model interactions of two AC voltage sources connected by means of an RL passive element, and the third equation represents voltage balance on the DC filter capacitor. Note that d and q channels are cross-coupled. Voltage vrd1 is

obtained by balancing currents ird1and igd1 on a local very large resistive load

(RD), situated at PCC, same remark being applicable to channel q.

Equations (1) may be completed with the DC line dynamic, given by equa-tion (2). Here, LDC and rDC are the inductance and the resistance of the

high-voltage DC line, respectively and vDC2 is DC voltage at station S2, see

HVDC link layout in Fig. 2.

2LDCiDC˙ = vDC1− vDC2− 2rDC· iDC. (2)

Further, concerning the station control, its overall scope is to track imposed levels of both active and reactive power values, on d and q channels, respectively. In the present work, VSC classical control structure has been adopted, VSC controllers being developed in a new dq frame linked to the PCC1 by means of a phase locked loop (PLL), like in [15]. This basically requires a changing of dq coordinates from the infinite bus to a frame linked to PCC1; therefore, the estimation of the PCC1 angle is done by using a PLL which phase detector is the so-called Kron transform – see [17, 10]. The new variables expressed in the new (PCC-linked) dq frame will be denoted by idq1, βdq1 and vdq1 (for

station S1). Relation with variables from equation (1) is given by the Kron transform – see equations (3) for the case of currents and duty ratios:

       id1 = ird1· cos δ1− irq1· sin δ1 iq1 = ird1· sin δ1+ irq1· cos δ1 βrd = βd· cos (−δ1) − βq· sin (−δ1) βrq = βd· sin (−δ1) + βq· cos (−δ1) , (3)

with δ1 being the phase shift between the two frames.

Using measurements from the station S1 (PCC1) (currents and voltages from d and q axis in PLL frame) one may compute active and reactive power, respectively as:



P1 = 1.5 (vd1· id1+ vq1· iq1)

Q1 = 1.5 (vd1· iq1− vq1· id1) .

(4) The classic control structure employs inner current control loops with cross-decoupling d−q structure that use proportional-integral laws – see, for example, [16]. Current components references are provided by the outer control loops.

On q channel, the outer loop aims at tracking a composite variable repre-senting a mix between the PCC1 voltage amplitude, vP CC and reactive power,

Q1. This variable, introduced in [18], is denoted in the sequel by M1, and is

given by equation (5):

M1= vP CC1+ λ · Q1, (5)

λ being the reactive power coefficient (measured in kV/kVar) and PCC voltage amplitude is vP CC1= q v2 d1+ v 2 q1.

It can be shown that the plant to be controlled can be expressed as a gain, therefore a simple integral controller can be used for PCC voltage amplitude and for reactive power, as in [16].

On d channel, either active power at PCC or DC-link voltage can be con-trolled within the corresponding outer loop. VSC of station S1 is concon-trolled in active power, HVDC link DC voltage being ensured by station S2. In this case the plant can be easily approximated with a gain, which is PCC voltage amplitude – see first equation of set (4), by considering vq1 ≈ 0. In this case

also one usually employs an integral controller.

The overall control law is expressed in dq frame linked to PCC and is given by equations set (6) – see [16]:

       βd1 = −Kpc(1 + 1Tics) (i∗d1Tics + 1 − id1) + 2ωLr1vDC1· iq1 βq1 = −Kpc(1 + 1Tics) i∗q1Tics + 1 − iq1
− 2ωLr1vDC1· id1 i∗ d1 = 1TiPs · (P1∗− P1) i∗_q1 = 1TiMs · (M1∗− M1) , (6)

where s is the Laplace operator and variables h·∗i denote the associated set-points. Kpc and Tic are the proportional gain and integrator constant of PI

current controllers, TiP is the active power controller integrator constant and

TiM is the reactive power controller integrator constant. βd1 and βq1are

Note that, in real-world application, a generic PCC phase angle is extracted from its voltages by using a three-phased PLL system and all computations in the station control system are done in the coordinates linked to the PLL angle. It is known that the PLL synchronizes on PCC angle with a certain dynamic (depending on its internal controller parameters), its actual value being delayed from the PCC angle in dynamic regime. Thus, it induces a non negligible effect on the overall dynamic behavior of the controlled VSC, negatively impacting its stability – see [19]. Here, the adopted PLL structure is the one represented in Fig. 3. One may show that, by using Kron transform as a phase detector, this structure outputs the same angle as a 1st _{order classical PLL and exhibits}

the same dynamics. In short, in steady state this loop outputs the necessary value of angle δ that nullifies the voltage vq1; this angle represents the phase

shift between the PCC-linked and infinite bus-linked dq frames.

2π E 1 s PI controller Σ 0 + − Kron transform Σ ₋ + δ₁ !t v_rd1 v_rq1 v_d1v_q1

Figure 3: PLL structure using Kron transform as phase detector (station S1).

PLL dynamics intervene twice: in id1 and iq1 computation in direct Kron

transform (i.e., in variables measurement) and in βrd1 and βrq1 computation

(actuation in inverse Kron transform), by means of angle δ1with a non-negligible

influence in the overall closed-loop dynamic behavior. Its modeling is the same for all four stations in the benchmark in Fig. 1.

Concerning station S2, the plant is modeled similarly as in the case of station S1, by using equations (1). The inner control loop is the same, but the outer loop is different: on q channel, its target is to track M∗

2 variable, similarly as

station S1; on d channel its target is different – to regulate the HVDC line voltage at the desired value.

Note that the third equation of set (6) is valid only for the power-controlled station/VSC, which is the case of station S1. Station S2, which uses voltage control, employs equation (7) instead – see [20].

i∗_d2= Kpv(1 + 1Tivs) · (v∗DC2− vDC2) (7)

Therefore, in HVDC1 one imposes active power value to station S1 to be injected/drained from the grid; station S2 will drain/inject the required amount of active power necessary to balance the HVDC voltage. The control system of a single HVDC line has been tuned in order to obtain pertinent performances for id and iq: damping at around 0.65 - 0.85 and time constant 5 ms for the inner

corresponds to ≈ 17 ms for vDC control loop and to 20 - 30 ms for active power

P and variable M . All these outer loops have variable damping depending on grid equivalent short-circuit ratio (SCR) at the corresponding station, (i.e., the AC impedance) and on the operating point. SCR value is classically defined as in [21]:

SCR = SACPDCr,

where SACis station bus (PCC) short-circuit capacity and PDCris HVDC rated

power.

2.3

Benchmark modeling

Further, the second HVDC in Fig. 1 is modeled in the same manner as the HVDC in Fig. 2 and detailed in the section above. Similarly, station S3 is power-controlled and station S4 is voltage-controlled.

ConAC line is modeled as a RL branch, by using equations similar to equa-tions (4) and (5) from equation set (1).

The synchronous machine model employed in this paper is the classical one, expressed in rotor-linked dq frame as given, e.g., in [22] (step-up connexion transformer being ideal). A classical control structure, containing two loops, one for terminal voltage control (excitation) and one for output power control (turbine-generator governor) has been employed [22]. The parameters from the control structure have been chosen following classic and best engineering practices, in order to match the following target dynamic performance (in terms of power): rise time of 25-30 s with a suitable damping (e.g., larger than 0.2, for the operating range 10%-100% of rated power. It has been further tested on an equivalent grid including an infinite bus and AC line lengths between 50 and 400 km.

3

A mixed analysis approach

The setup in Fig. 1 has been implemented in Simulink R _{toolbox of the MATLAB} R

software by means of nonlinear modeling detailed in Section 2. On this model, extensive simulations have been performed in order to emphasize the behavior of the two coupled HVDC links and to make a preliminary assessment of their interaction.

In a second step, system linearization and modal analysis are targeted in order to better quantify the HVDCs interaction. As a preliminary step, all system states (integrators) in the Simulink R _{diagram are labeled with suitable}

identifiers (e.g., ”int PI iq3“ means the state associated with the iq3 current

controller integrator). Then, after eliminating nonessential nonlinearities (satu-rations, etc), linearization is done numerically, by using the Linear analysis tool available in the above-cited software around a steady-state operating point, cor-responding to a certain scenario. For consistency, in all scenarios the input is chosen as M₁∗(the setpoint) and the output as M1, although any of variables Pk

4 4.2 4.4 4.6 4.8 5 1.9 1.95 2 2.05 2.1 2.15 2.2 2.25 2.3 ×10 5 a) M1 [kV] t [s] 4 4.2 4.4 4.6 4.8 5 1.9 1.92 1.94 1.96 1.98 2 ×10 5 b) M3[kV] t [s] 4 4.1 4.2 4.3 4.4 4.5 1.9 1.95 2 2.05 2.1 2.15 2.2 2.25 ×10 5 c) M1, M3 [kV] t [s] 4 4.2 4.4 4.6 4.8 5 1.9 1.92 1.94 1.96 1.98 2 ×10 5 d) M3[kV] t [s]

Figure 4: Nonlinear system response to a M1∗ step: a) and b): nonlinear system

response at M1 stepoint step variation, for various ConAC line lengths black –

500 km, blue – 200 km, red – 100 km; c): detail of M1 (black) and M3 (red)

evolutions at a M1 setpoint step, where ConAC length is LAC =100 km; d):

M3evolution at M1stepoint step variation, for various SCR values (black – 3.5,

blue – 5, red – 7.5).

and Mk with k = 1..4, may be set as output. The linearization outputs system

state matrix, A, together with a vector of state variables labels.

Modal analysis for continuous-time linear time-invariant system starts with the state matrix. Based on this matrix, normalized right eigenvectors vi and

left eigenvectors wi, which verify



A · vi = vi· λi

wi· A = λi· wi

, (8)

with wi· vj = 0 for i 6= j and wi· vj = 1 for i = j – see [11] – are determined,

The participation factor of variable xjin mode i is computed as the product

between elements j of vectors wi and vi:

pji= (wi)j· (vi)j, (9)

see [11] for quick reference.

One determines participation factors for all state variables and for all system modes, i.e., the so-called participation factor matrix (square matrix of system dimension) – see [11]. Absolute values of participation factors are further used, to compare the influence of variables that intervene in a certain mode. All oscillatory modes are analyzed in order to emphasize the systems’ shape of oscillations. The ones poorly-damped and related to electrical couplings (i.e., which involve distant devices) are of particular interest.

Results concerning this analysis will be presented in the next sections by means of scenarios that can be split into two categories: with or without G1 generator connected in the AC grid. Irrespective of the situation, only pa-rameters and setpoints of stations S1 and S3 will change; stations S2 and S4 setpoints are voltages v∗_DC2= v∗_DC4= 640 kV and M2∗= M4∗= 332 kV (rated

values). Their associated SCR is always constant, at ≈ 5. The values consid-ered for ConAC line length correspond to the initially-stated “close” HVDC lines context. Setpoints of the synchronous generator are constant determining a steady state at around Pe= −400 MW, in terms of active power and at about

Qe = −100 MVAR, reactive power. Both high-voltage DC lines have a length

of 200 km. In this paper passive sign (load) convention has been used.

4

Coupling mode identification by nonlinear

sim-ulation

4.1

Time-domain numerical simulations.

In order to make a preliminary assessment of the coupling behavior, one choses the structure without generator G1, in which the operating context consists in injecting into the AC grid active power values P1 = −500 MW and P3 =

−250 MW and in maintaining M3= 332 kV (at rated value); also, both S1 and

S3 have the same SCR=5 value. Simulations on the nonlinear Simulink R _model

have been performed and a large step of M1 setpoint, has been imposed.

Fig. 4 a) presents M1 variable evolutions (local control loop voltage tracks

setpoint variation), and Fig. 4 b) M3evolutions – corresponding control loop

re-jects the perturbation, both for various lengths of interconnection line, ConAC. System’s dynamic behavior has two main components: a damped behavior whose frequency becomes smaller as ConAC line length decreases, and a second undamped behavior, whose frequency is around 20 Hz (see also Table 1) and which is an oscillating mode that couples the two HVDCs. Indeed, in Fig. 4 c) a detail of M1 (black) and M3 (red) evolutions at a M1 setpoint step (ConAC

3.9 4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 1.91 1.9105 1.911 1.9115 1.912 1.9125 1.913 1.9135 1.914 10 5 Setpoint No feedforward Feedforward ( Vd only) Feedforward w/ filter (666 rad/s)

M1[kV]

t [s]

Figure 5: Nonlinear system response to a M₁∗ step: black line – set-point evo-lution, red line – case without feedforward, blue line – case with feedforward, magenta line – case with filtered feedforward.

outputs – oscillations have the same frequency, the same damping and the same phase, suggesting that the two input-output transfer channels are coupled, as their oscillations are due to the same mode.

4.2

Why not using feedforward component in current loops?

An alternative common approach is to use a feedforward component in the low level control loops to obtain better behavior in terms of stability. This is indeed useful for the case of a weak grid, where PCC voltage varies significantly – i.e., vd1 in 2nd equation of (1). The feedforward is done by measuring vd1 voltage

and reinjecting it into the duty cycle control law, i.e., in 2ndequation from (6). For sake of comparison, this alternative solution has also been implemented in S1-S4 controls and Simulink R _{numerical nonlinear simulations have been}

performed.

One observes in Fig. 5 that the voltage response is totally smoothed – blue line in the case where ideal PCC voltages measuring would be available. This means that the oscillatory modes are not present anymore (or are sufficiently damped) and the feedforward action is effective. However, in real-world applica-tions, the measure of PCC voltage is affected by filtering. When the feedforward component uses the filtered value of PCC voltage, the system response becomes oscillatory, again – see the magenta line in Fig. 5. So the new dynamics intro-duced by the filtered PCC voltages appear to counteract the effect of a genuine feed-forward and thus the increase in the system complexity is unjustified.

Further, the analysis will be done without the feedfordward components in the low-level control loops – i.e., as in Subsection 4.1, by using equations (6), for sake of being coherent with an analysis/design of the system in its totality, free of local and only partially effective control “patches”.

5

Modal analysis of HVDCs interaction

5.1

Parametric analysis for various operating scenarios

For the above-described context, the nonlinear system is linearized around the associated steady-state operating point, by using numerical tool in Simulink R

software; thus, a 62-states linear model is obtained. Modal analysis algorithm – based upon equations (8) and (9) – is applied and the previously-observed oscillatory mode (see Fig. 4), which will be further referred to as Mode 29, is identified. Its frequency is around 20 Hz and it is poorly damped in some situations which will be discussed next. Most participative states in Mode 29 are the ones contained into the inner control loops (PLL and current controllers), see also Table 2. Variables belonging to S1 – from the HVDC link 1 – and S3 – from the HVDC link 2 – have the most important contributions to Mode 29. This confirms the conclusions of the previous time-domain analysis, that the two HVDCs interact.

Next, variations of Mode 29 parameters with respect to the operating context will be studied.

The first scenario involves modal analysis for three different SCR values at stations S1 and S3. Steady-state active power values are P1 = −500 MW

and P3= −250 MW, M variables M1= M3= 332 kV and ConAC line length

is LAC=150 km.

Fig. 4 d) shows M3 evolution (non-linear system simulations) at a step on

M1setpoint. Note that the system is less damped as SCR values at both S1 and

S3 are smaller (the grid is weaker). This can also be seen in the modal analysis results. Table 1 (left side) shows Mode 29 main parameters for the three SCR values. Frequency has not an important variation, but the damping is severely decreasing with SCR, thus affecting the system stability.

Fig. 6 a), which contains the system pole’s evolution as SCR decreases, shows the same trend. Further reduction of SCR values makes the system unstable. Table 2 contains ranking of most participative system states in Mode 29 for two values of SCR. Note that most participative states belong to inner loop controllers and to PLLs of either station S1 and S3.

The second scenario concerns modal analysis for various active power values (positive/power draw or negative/power injection) required for stations S1 and S3. The set-up presumes a ConAC line of 150 km, both S1 and S3 maintain variables M1 and M3 at their rated values (i.e., 332 kV); SCR is the

same for S1 and S2, i.e., 3.5. Stations S2 and S4 follow operation of stations S1 and S3 respectively, by regulating DC voltage at the rated value (640 kV).

Table 1 (right side) shows Mode 29 parameters variation with P1 and P3

values. Damping of Mode 29 is lowest when both S1 and S3 inject power in AC grid. This is also visible on Fig. 6 b), showing variation of pole associated with Mode 29 with respect to power levels. Ranking of first nine states participating in Mode 29 are shown in Table 3. The inner loop (current) controllers states, including PLLs (either the PLL integrator or its PI controller), have the most important participation in this mode. Note also, that these states belong to

-160 -140 -120 -100 -80 -60 -40 -20 0 -300 -200 -100 0 100 200 300 a) Im Re Mode 29 -30 -25 -20 -15 -10 -5 0 -150 -100 -50 0 50 100 150 b) Im Re Mode 29

Figure 6: a) System pole-zero maps for various SCR values: black – 5, blue – 4, red – 3; b) System pole-zero maps for various P1 and P3 setpoints; red:

P1 = −500 MW, P3 = − 600 MW, blue: P1 = −500 MW, P3 = +600 MW,

magenta: P1= +500 MW, P3= −600 MW, black P1= +500 MW, P3= +600

MW.

stations that inject active power. For example, ranking in column 3 of Table 3, corresponds to case where S1 draws power and S3 injects power from/in their AC grids. Therefore, S2 injects power and S4 draws power in/from their AC grids in order to maintain DC voltage at the prescribed value. States belonging to stations S2 and S3 are the most participative. Table 3 shows that this remark holds for all analyzed cases.

Notice that the coupling mode is systematically better damped in scenarios where power-controlled stations (S1 and S3) draw power – see the right side of Table 1. Also, note that in all cases the states belonging to the ConAC line do not significantly intervene in Mode 29. Also, states belonging to VSC reactors, AC lines and low-level loop prefilters – see equations (6) – may also be significantly involved in the participation mix.

5.2

Modal sensitivity analysis wrt. to ConAC line length

This subsection aims at quantifying the influence of the interconnection line length, LACon the interaction Mode 29, based on results presented in [24] and

[25]. From the definition of sensitivity one can show that variation of eigenvalues λi of system matrix A, wrt. to parameter LACmay be expressed as in equation

(10):

SLi= ∂λi∂LAC=

X

kj

∂λi∂akl· ∂akl∂LAC, (10)

for all i = 1..n, n being the system order and akl the element of line k and

column l of matrix A.

A single term from equation (10), further called partial sensitivity skl, may

be written as:

skl = ∂λi∂akl· ∂akl∂LAC = wikvil· ∂akl∂LAC, (11)

with k, l = 1..n, v and w the right and left eigenvectors of λ, respectively. Therefore, the sum of all elements described by equation (11) represents the total sensitivity of a certain mode with respect to the parameter LAC.

Further, equation (11) is used to estimate the elements of variable SL29, i.e.,

sensitivity of Mode 29 wrt. to ConAC length; this is done by employing two system configurations, each of them having different ConAC line length. The procedure runs as detailed bellow.

#1. First, one obtains linearized model (and hence matrix A1, eigenvectors v and

w) for a ConAC length LAC = 100 km. Then, the procedure is repeated for a

longer line of LAC= 200 km and one obtains matrix A2.

#2. One easily computes elements ∆akl

∆LAC (an estimate of the last component in equation 11), by subtracting matrix A1 from A2, ∆A = A2− A1, and obtaining

variables ∆akl, for all k, l = 1..n.

#3. Only the significant values ∆akl

∆LAC, implying an important dependence with the ConAC length, are further used (all other are discarded). One obtains a vector containing this variables by sweeping matrix ∆A line by line. One also gets a vector with line indices, k and another one with column indices, l corresponding to the preserved values.

#4. The selected mode being i = 29, one can also select eigenvectors v29land w29k,

by sweeping vectors of column indices and of line indices obtained at step #3. Then, one computes partial sensitivities sklfor all the significant values _∆L∆akl

AC, by using equation (11), and A1 eigenvectors.

#5. Summation of all partial sensitivities computed at step #4 gives the Mode 29 sensitivity wrt. ConAC line length.

Further, one studies this sensitivity when the system has a symmetrical structure and when it has a certain asymmetry. Two asymmetrical cases have been analyzed. First, the asymmetry is given by different SCR values at stations S1 and S3. Secondly, the asymmetry is functional (due to power flow), active power values at S1 and S3 are different, i.e., ConAC line is charged.

5.2.1 Scenario 1: symmetric structure

Both active power levels, P1 and P3 are −500 MW (injected into AC grid).

First, one considers both infinite buses AC1 and AC3 with the same SCR=3.5. Fig. 7 a) shows the partial sensitivities, for this case. Note that only six values

∆akl

∆LAC have been taken into account (the others are insignificant); j is the index

in ∆akl

∆LAC (or else in vectors k and l). The total sensitivity value is very small

SL29 ≈ 3.4e − 6, which suggests quasi invariance of Mode 29 parameters (and

associated participation factors) with ConAC line length. 5.2.2 Scenario 2: structural asymmetry

1 2 3 4 5 6 0 0.5 1 1.5 2 2.5×10 -6 a) skl j 0 10 20 30 40 50 60 70 0 50 100 150 200 250 300 350 400 b) skl j

Figure 7: Partial sensitivities for: a) symmetric case (equal SCRs at AC1 and AC3); b) structural asymmetric case (AC3 has 10% smaller SCR than AC1).

In a second case, one considers a small difference between SCR values in infinite buses AC1 and AC3; the first one is 3.5, the latter being 10% smaller. Fig. 7 b) shows the magnitude of sensitivities for this second case. This case

has 62 significant values that have been retained. Note that in this second case partial sensitivities are significantly larger wrt. the first case; total sensitivity is SL29≈ 659.4.

To conclude, as the system is symmetric (not only from the structure point of view, but also from the power flow point of view), the sensitivity of Mode 29 wrt. ConAC line length is extremely small. However, even a small imbalance in system structure determines a significant sensitivity of the studied mode wrt. LAC.

5.2.3 Scenario 3: asymmetry in power flow

This case considers symmetrical system structure wrt. middle point of ConAC line. This means that SCR values are equal for S1 (AC line 1) and S3 (AC line 3); this is also valid for S2 and S4. However, active power levels imposed to each of stations S1 and S3 differ. Table 4 shows the output of the above-detailed procedure for four different power flow situations. Likewise in the previous scenario, in the asymmetrical cases (when ConAC line is loaded) the total sensitivity of Mode 29 wrt. ConAC length is significantly higher than in the symmetrical ones. Fig. 8 shows the distribution of partial sensitivities for the asymmetrical power flows. In this cases it is expected that parameters of this mode have important variations with the line length variation. This analysis also implies that, in the symmetric cases, Mode 29 is insensitive to this parameter.

6

Minimal modeling to capture the HVDC links

interactions

Analysis above was carried out with a very detailed modeling in order to be sure not to lose any relevant dynamics. The question is now to state which are the minimal modeling requirements to capture the coupling mode put into evidence between the two HVDCs. This is done to further facilitate large-scale AC grid studies. For this, separate influence of certain dynamics over the HVDCs interaction, are analyzed. Scenarios considered next have: ConAC of 150 km, SCR1=3.5 and SCR3 = 3 (which makes a slight structural unbalance between S1 and S3). Steady-state operating point is imposed by P1= +500 MW

and P3= −600 MW in order to have operational asymmetry. Steps of reference

voltage are applied at S1 (M∗

1 is the input, M1 is the system output).

6.1

System containing ConAC line without dynamics

As it has been recalled before, RL-based ConAC line model is given by equations four and five of set (1), in the form of equations (12):



Lcicd˙ = ωLc· icq+ vrd1− vrd3− rc· icd

Lcicq˙ = −ωLc· icd+ vrq1− vrq3− rc· icq

0 10 20 30 40 50 60 70 80 0 500 1000 1500 2000 2500 3000 3500 4000 a) skl j 0 10 20 30 40 50 60 70 0 500 1000 1500 2000 2500 3000 3500 4000 b) skl j

Figure 8: Partial sensitivities for: a) P1 = −500 MW, P3 = +500 MW; b)

P1= +500 MW, P3= −500 MW.

where icd and icq are ConAC line current components, vdr1 and vqr1 are the

PCC1 voltage components at station S1 and vdr3 and vqr3 the PCC3 voltage

components at station S3. Lc and rc are the inductance and the resistance of

ConAC, respectively.

If one nullifies the derivatives on both d and q axis and neglects the line resistive effect, one obtains an algebraic model of the ConAC line, yielding its current components as:

 _i

dc = _ωL1_c(vqr1− vqr3)

iqc = −_ωL1_c(vdr1− vdr3) .

(13) Further, the benchmark in Fig. 1 is implemented by using ConAC model given by equations (13). As previously, one linearizes (by using numerical tool in Simulink R_{) system around the corresponding steady-state operating point}

Fig. 9 a) shows the step response of the newly-obtained, reduced order system, in red vs the one of the full-order original system, described by the black trace. Note that the response is quite similar, the reduced-order system also has an undamped response of about the same frequency and damping. This observation is backed by the pole maps in Fig. 9 b). The concerned pole pair, situated at about 20 Hz, is perfectly superposed over the original system with full dynamics (see red and black crosses).

0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 a) M1∗→ M1 t [s] -70 -60 -50 -40 -30 -20 -10 0 10 -150 -100 -50 0 50 100 150 b) Im Re   Superposed poles

Figure 9: Black – system with ConAC dynamics, red – system without ConAC dynamics; a) Step responses; b) Pole maps.

Finally, full modal analysis is done on the newly-created, reduced-order sys-tem. It has been noted that Mode 29 frequency and damping are not sensibly affected by ignoring ConAC line dynamics. Also, ranking of the most impor-tant states influencing Mode 29, and their associated participation factors are the same as in the full-order case.

The above observations gathered from the step responses, pole maps and modal analysis allow us to state that the dynamics of ConAC line, intercon-necting the two HVDC lines do not significantly affect the interconnection Mode

29. Therefore, simulations of large power systems that include HVDCs would not require complete dynamic modeling of AC lines. Static models that em-ploy equations (13) are sufficient in order to properly capture the behavior of interconnected HVDC lines. This will significantly reduce the computation bur-den associated with the simulation of large power systems that embody a large number of lines.

6.2

System with instantaneous inner (current) VSC loop

dynamics

Here, the low-level inner control loops for id and iq currents are reduced to

instantaneous current sources. As previously, the effect of this model reduction will be assessed vs the initial full-order model.

To this end, in the benchmark containing HVDC interacting links one uses directly the variables i∗_dk and i∗_qk, with k = 1, ..., 4 (current references for all HVDC stations) both in the AC lines equations and in the ones that give the active and reactive power values Pk and Qk. PLL dynamics are taken into

account here, as variables i∗_dkand i∗_qk need coordinate (Kron) transform. In this case, dynamic interaction between S1 and S2 and between S3 and S4 are completely decoupled: currents idqat station S1 do not depend anymore on

the vDC voltage, which is a variable shared with S2; hence, this later (given by

the model of S2) does not influence dynamics at station S1. This implies that HVDCs interaction is reduced only to interaction between models of stations S1 and S3.

If the model reduction applies only to station S1, i.e., i1d≡ i∗1dand i1q ≡ i∗1q,

a 38thorder system is obtained by using the same procedure as in previous cases. Fig. 10 a) gives the step response of the reduced-order linearized system (in blue line) plotted vs the one of the original system (plotted with red line). Note that the reduced-order system is much more damped. The same effect is seen on Fig. 10 b) which provides the pole maps of reduced-order system (blue crosses) vs the one of the original system (with red crosses). Note that many of the poles, including the pole of interest, are missing from the reduced-order system.

Table 5 shows two modes of the reduced-order linearized system in the fre-quency range of interest. The mode closest in frefre-quency to the Mode 29 of the full-order system is one of 19,89 Hz, in which participate mostly states from inner loops associated to stations S3 and S4. Thus, this is not a coupling mode via the grid.

The above-presented analysis clearly shows that the system simplification by neglecting low-level current loops yields a reduced-order system which does not retain any longer specific behavior of the two interacting HVDC lines, analyzed in previous sections of this report. The mode of interest (Mode 29 ) is no more present in the reduced-order system, the overall response is more damped, the system is farther from instability wrt. the original, full-order case.

6.3

System with instantaneous PLL dynamics

In this section PLL dynamic behavior is neglected. The PCC angle is directly computed in numerical simulation by using equations set (1). For station S1, these equations output voltage components at PCC1 as vd1 ≡ vrd1 and vq1 ≡

vrq1and one may compute the PCC1 voltage angle as:

δ1= − arctan

 vq1

vd1

 .

This angle is then used in Kron transform to obtain voltage and current images in the PLL dq frame to be used in control algorithm.

The nonlinear system step response with such an ideal PLL for station S1 is given in Fig. 11 a), with blue line. Note that the response is significantly more damped wrt. the case of the full-order original system. Fig. 11 b) shows the

0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 a) M₁∗→ M1 t [s] -70 -60 -50 -40 -30 -20 -10 0 10 -150 -100 -50 0 50 100 150 b) Im Re   Initial mode 29

Figure 10: Red – system with idqdynamics, black – system without idqdynamics

displacement of the oscillatory pole in the complex plane – see the blue crosses wrt. the red crosses (the full-order model). The measured damping is 0.088.

4 4.2 4.4 4.6 4.8 5 1.91 1.9105 1.911 1.9115 1.912 1.9125 1.913 1.9135 1.914×10 5 4 4.1 4.2 1.911 1.912 1.913 ×105 a) M1 [MW] t [s] -70 -60 -50 -40 -30 -20 -10 0 10 -150 -100 -50 0 50 100 150 b) Im Re 19.61 Hz 0.026 22.31 Hz 0.088

Figure 11: Red – system with PLL dynamics, black – system without PLL dynamics on S1 only; a) Step responses; b) Pole maps.

Table 6 comparatively shows ranking of states involved in the mode of in-terest. The case of the full-order system is in the left side and the case with instantaneous PLL at station S1 is in the left side. Note that frequency and damping of the mode differ significantly. Ranking of states and participation factors in Mode 29 differ significantly to the point that this mode involves only variables of station S1, i.e., it is no longer a coupling mode between the two HVDCs.

To conclude, the above presented analysis clearly shows that the system simplification by neglecting the PLL dynamics yield a reduced-order system that does not retain any longer the specific behavior of the two interacting HVDC lines, analyzed in previous sections of this report. The Mode 29 of interest is significantly shifted in the reduced-order system (depending of the number of neglected PLLs), the overall response is more damped, the system is farther

from instability wrt. the original, full-order case.

7

Analysis in the case of AC grid with synchronous

generator

Next, a controlled synchronous generator has been inserted in the dual HVDC benchmark, as shown in Fig. 1. In the new structure one actually splits AC1 impedance into two equivalent RL lines, in order to obtain sound comparison base with the previous case. The synchronous generator is inserted closer to station S1 than the infinite bus such that AC line 6 is about 50 km and AC line 5 is 150 km.

The target here is to apply the mixed approached described in Section 3 to the entire system in Fig. 1, including not only the two HVDC links, but also the synchronous generator. Basically, all the operations and analysis made on the previous case are repeated for this new set-up. ConAC length is 150 km. SCR1=3.5 and SCR3=3 (which makes a slight structural unbalance between S1 and S3). Setpoint voltages at Stations S1-S4 are at their rated values. Steady-state operating point corresponds to P1 = −500 MW and P3 = +600 MW.

Steps of reference voltage, M₁∗, are applied at S1 for observing the overall system dynamic behavior.

7.1

Interaction mode identification

7.1.1 Case 1 – positive M₁∗ step of 10% of rated value of M1

Fig. 12 shows the overall system (i.e., the one containing also synchronous generator G1) behavior, with red color vs the one formed only by the two interacting HVDC links, with black color. The focus is here on the variables belonging to HVDC stations. The new system is not significantly faster than the old one. Moreover, one observes that the system including generator G1 behavior is less damped wrt. the original one, the system has even smaller stability reserve. In this new case, the coupling oscillatory mode has frequency of ≈ 18 Hz with damping smaller than 10%. Supplementary simulations show that these parameters depend on the steady-state operating point (signs and values of P1, P3 and Pevariable).

7.1.2 Case 2 – negative M₁∗ step of 5% of rated value of M1

Fig. 13 shows the system response, the focus being here on the generator behav-ior. Note that the oscillatory mode can be observed in the generator’s power, Pe: it is the spike at moment 50 s. The other lower frequency mode is about

0.25 Hz.

These time-domain simulations have shown that overall system dynamics are changing when a synchronous generator is inserted in the AC grid (in this case within the line AC1). So, a new modal analysis will be done in this section,

49.9 50 50.1 50.2 50.3 50.4 50.5 50.6 50.7 50.8 50.9 1.8 1.85 1.9 1.95 2 2.05 2.1 2.15 2.2 2.25 2.3×10 5 a) M1[V] t [s] 49.9 50 50.1 50.2 50.3 50.4 50.5 50.6 50.7 50.8 50.9 1.8 1.85 1.9 1.95 2 2.05 2.1×10 5 b) M3[V] t [s]

Figure 12: System response to a voltage reference step at S1: red – interact-ing HVDC links includinteract-ing generator G1; black – interactinteract-ing HVDCs without generator G1. Evolutions of: a) S1 voltage, M1; b) S3 voltage, M3.

in order to full assess the behavior of this system. The main target is the high-frequency mode (situated in the neighborhood of 20 Hz), which is similar to Mode 29 of the original system and which will be further denoted as Mode 38.

7.2

Modal analysis

Liniarization around a certain operating point has been done numerically for each subsequent scenarios and a new linear model with 72 states, whose param-eters depend on the current scenario, is obtained.

Table 7 shows Mode 38 parameters – in the case with generator, at the left side – vs Mode 29 parameters, without generator – at the right side, for various active power flows. Note that generator currents have important participation factors, and mode parameters (frequency and damping) significantly differ. But new frequency is not much lower, 18.5 Hz vs 20.5 Hz, but the reduction in

40 50 60 70 80 90 100 110 120 130 140 150 -4.15 -4.1 -4.05 -4 -3.95 -3.9×10 8 a) Pe[W] t [s]

Figure 13: output generator power Peresponse to a voltage reference step at S1

damping is very important (i.e., from 8% to 0.6%). Basically, one observes here the same phenomenon like in the previous case of HVDCs interaction without synchronous generator – see Sections 4 and 5.

Also, insertion of the generator breaks the link between the participating stations and the sign of the power flow, observed in the previous case (without generator). In this sense, Table 7 shows that station S4 has been excluded from the participation top ranking; its states have been replaced by ones belonging to the generator.

Therefore, synchronous generator insertion in the benchmark containing the two HVDC links changes the interaction behavior, and modifies the parameters of the main oscillatory mode. The frequency becomes slightly lower (e.g., from 20 Hz to 18 Hz) and the damping becomes smaller. Some operating regimes (at high level of active power values or in the case where both S1 and S3 inject active power) become unstable, which reduces system effective operating range.

Acknowledgment

The authors would like to thank Dr. Hani Saad from R´eseau de Transport d’Electrict´e (RTE) de France for his useful remarks on the manuscript.

8

Conclusion

This paper has dealt with analysis of two close-connected HVDC lines by using high-fidelity models in dq frame.

A potentially inconvenient interaction, due to HVDCs coupling, has been highlighted through time-domain simulation. Modal analysis has shown that this interaction consists in an oscillatory (i.e., complex) mode between convert-ers of distant HVDCs. The low-level controllconvert-ers and PLLs states have the

high-est participations in this coupling oscillatory mode. A quite exhaustive study of the variation of the parameters of the system, as well as the modification of the control structure, have shown that this mode exists in all situations, i.e., it is related to a structural coupling phenomena. Only its damping varies with the parameters of the system (it is smaller as AC grid SCRs decrease) and, in gen-eral, depends on the power flow. It may be under 5% if both AC-linked stations inject active power into the grid, or if synchronous generators are present in the system. It has also been shown that this coupling phenomena can be accurately captured with transient-stability models, i.e., neither dynamic models for the AC lines, nor high-frequency (to capture the switchings) representation of the converters are necessary. This significantly facilitates the analysis in a realistic (large-scale) grid context, as needed by TSOs to state the stability and coor-dination of controls. Of course, high-order harmonics aspect should be studied apart, with higher-frequency models.

To some extent, this kind of interaction mode is similar to the well-known inter-area modes existing between rotating generators.

Even if the study is done on a system with simple topology, the main char-acteristics of real situations are captured by the AC lines which connect the two HVDCs and the infinite buses. This models the electrical distance on a real power system and it is expected that the conclusions above hold in the general case.

The way in which the control should be adapted to damp this kind of modes in critical situations is now under development and will be presented in forth-coming publications.

9

Symbols used

int PI iq1 – iq PI controller at S1 – see 2nd eq. of (6)

int PLL1 – integrator of phase locked loop at S1 – see Fig. 3 int PI id3 – idPI controller at S3 – see 1st eq. of (6)

Reactor1/int iq – iq current of reactor at S1 – see 2nd eq. of (1)

Reactor 3/int iq – iq current of reactor at S3 – see 2nd eq. of (1)

int PI id4 – idPI controller at S3 – see 1st eq. of (6)

int PLL4 – integrator of phase locked loop at S4 – see Fig. 3 int PI PLL1 – integrator of PI controller of PLL at S1 – see Fig. 3 AC line 1/int iq – iq of line AC1 (linked to S1) – see 5th eq. of (1)

AC line 4/int id – id of line AC4 (linked to S4)– see 4th eq. of (1)

Prefilter id2 – prefilter for id PI controller at S2 – see 1st eq. of (6)

int PI vdc4 – integrator of vDC PI controller at S2 – see eq. (7)

DC filter 3/int 1 – state of HVDC capacitor – see 3rd eq. of (1) int id – integrator of AC generator’s stator current equation, id

int if – integrator of AC generator’s field current equation if

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Table 1: Mode 29 parameters variations: left side – with SCR values, right side – with P1and P3 values.

SCR Freq. [Hz] Damp. P1, P3[MW] Freq. [Hz] Damp.

3 19.6 0.034 -500, -600 19.57 0.021 4 21.6 0.096 -500, +600 21.5 0.08 5 22.94 0.131 +500, -600 21.27 0.084 - - - +500, +600 22.08 0.081

A

Appendix: System parameters

• AC system: frequency 50 Hz, rated RMS voltage 400 kV, • AC line: resistance 0.03 Ω/km, inductance 0.79 mH/km,

• Transformer: secondary RMS voltage 230 kV, ratio 1.74, Rpr= 0.4 Ω, Lpr= 38

mH, Rsec= 0.132 Ω, Lsec= 12.6 mH,

• Line reactor: impedance 25.3 mH, resistance 0.08 Ω,

• DC line: voltage 640 kV, capacitor 220 µF , resistance 13.9 mΩ/km, impedance 0.159 mH/km, length 200 km,

• VSCs: active power 1000 MW, reactive power Q ∈ {−500; +400} MVAr, maxi-mum active current ±4 kA, maximaxi-mum reactive current ±1.6 kA,

• Control parameters: λ=13e-6 V/VAr, Tic = 6.6 ms, Kpc=0.6e-4 A−1, Ktc =

5 · Ktp= 500, Kpv= 0.034 A/V, Tiv= 25 ms, Tim= 0.28 s, Tip= 7.04 ms,

• Synchronous generator:

Srated = 555 MVA, VT rated = 24 kV, ntr = 9.58 (AC system RMS 230 kV),

if rated = 3500 kA, p = 1, J = 0.74e6 kg · m2, Fr = 50 N·m·s, ra = 3.1 mΩ,

Ld= 4.98 mH, Lq= 4.845 mH, Laf d= 40 mH, rf = 71.5 mΩ Lf f d= 576.9 mH,

Rloc= 11 kΩ,

• Synchronous generator control:

Tp1= 10 s, Gp= 0.05, Tp2= 250 s, TmLim= 1.6e6 Nm, Tv1= 100 s, Tv2= 10.5

Table 2: States influences in Mode 29 for two values of SCR.

SCR → 5 SCR → 3

State name Participation State name Participation int PI iq1 0.09 int PLL1 0.15 int PLL1 0.09 int PI iq1 0.14 int PI iq3 0.09 int PLL3 0.13 int PLL3 0.08 int PI iq3 0.13 int PI id1 0.06 int PI id3 0.09 Reactor 1/int iq 0.07 int PI id1 0.1 Reactor 3/int iq 0.05 Reactor 1/int iq 0.09

int PI id4 0.05 Reactor 3/int iq 0.07 int PLL4 0.05 int PI PLL1 0.06 int PI iq4 0.05 AC line 1/int iq 0.05

Table 3: States with largest participation in Mode 29 for various P1 and P3

steady-state values.

P1, P3 [MW] P1, P3 [MW] P1, P3 [MW] P1, P3 [MW]

-500, -600 500,600 500, -600 -500, 600 State name Part. State name Part. State name Part. State name Part.

int PLL1 0.13 int PI iq4 0.07 int PI id2 0.17 int PI id4 0.14 int PLL3 0.13 int PI id4 0.07 int PLL2 0.15 int PLL4 0.12 int PI iq3 0.13 int PLL4 0.07 int PI iq2 0.14 int PI iq4 0.12 int PI iq1 0.12 int PI id2 0.06 int PLL3 0.1 Reactor4/int iq 0.09 int PI id3 0.1 int PLL2 0.06 int PI iq3 0.1 int PLL1 0.07 int PI id1 0.09 int PI iq2 0.05 Reactor2/int iq 0.1 int PI iq1 0.07 Reactor3/int iq 0.08 Reactor4/int iq 0.05 Reactor2/int id 0.09 Reactor4/int id 0.07 Reactor1/int iq 0.08 Reactor2/int iq 0.04 AC line2/int id 0.07 Prefilter id4 0.06 AC line3 /int iq 0.05 int PLL1 0.04 Prefilter id2 0.07 AC line4 /int id 0.06 int PI PLL3 0.05 int PI iq1 0.04 Reactor3/int iq 0.07 Reactor1/int iq 0.05 AC line1/int iq 0.05 int PLL3 0.04 int PI id3 0.07 int PI id1 0.05 int PI PLL1 0.05 int PI iq3 0.04 int PI iq1 0.06 int PI PLL4 0.04

Table 4: Total sensitivity of Mode 29 wrt. ConAC length for various active power levels.

P1, P3[MW] -500, -500 -500, +500 +500, -500 +500, +500

Table 5: Ranking of states for two oscillatory modes closed to the domain of interest of linearized reduced-order system in the case where only station S1 has no current dynamics.

Freq. 19,89 [Hz] – Damp. 0,292 Freq. 5,68 [Hz] – Damp. 0,314 int PI id4 int PI iq4 int PLL4 Reactor4/int id int PI id3 int PI iq3 AC line4/int id int PLL3 int PI PLL4 0,21 0,18 0,16 0,12 0,11 0,09 0,08 0,08 0,06 int PI vdc4 Prefilter id4 int PI id3 int P3 DC filter3/int2 DC filter3/int1 DC filter4/int2 DC filter4/int1 int PI id4 0,5 0,17 0,16 0,13 0,1 0,1 0,09 0,09 0,09

Table 6: Ranking of states for the interaction mode. Left side – the case of the original full-order system; right side – the case with neglected dynamics of PLL1.

PLL1– PLL4 with dynamics PLL1 without dynamics Freq. 19,61 [Hz] – Damp. 0,026 (Mode 29 ) Freq. 21,31 [Hz] – Damp. 0,088

int PLL3 int PI iq3 int PLL1 int PI iq1 int PI id3 int PI id1 Reactor 3/int iq Reactor 1/int iq AC line 3/int iq 0,13 0,13 0,13 0,12 0,1 0,09 0,09 0,08 0,05 int PI iq3 int PLL3 Reactor 3/int iq int PI id3 int PI PLL3 Reactor 4/int id Prefilter id4 AC line 3/int iq int PI id4 0,2 0,2 0,14 0,12 0,07 0,07 0,06 0,06 0,06

Table 7: Mode 29 parameters (left side) vs Mode 38 parameters (right side) – case where P1= −500, P3= +600 MW.

Freq. 21,6 [Hz] – Damp. 0,08 Freq. 18,41 [Hz] – Damp. 0,006 int PI id4 int PLL4 int PI iq4 int PLL1 Reactor 4/int iq int PI iq1 Reactor 4/int id Prefilter id4 Reactor 1/int iq AC line 4/int id 0,12 0,11 0,11 0,08 0,08 0,08 0,06 0,05 0,05 0,05 int id int if int PLL1 int PI iq1 int PI id1 Reactor 1/int iq int PI iq3 int PLL3 int PI PLL1 int M1 0,42 0,42 0,19 0,19 0,12 0,09 0,08 0,08 0,08 0,07